Given a simplicial complex L with vertex set I and a familyA = {(A(i), B(i))}i∈I of pairs of spaces with base points ∗i ∈ B(i), there is a definition of the “polyhedral product” AL of A with respect to L . Sometimes this is called a “generalized moment angle complex”. This note concerns two refinements to earlier work of the first author. First, when L is infinite, the definition of polyhedral ...

The Gale-Hoffman lemma characterizes feasible external flow (supply, demand) in a directed, capacitated network in terms of a number of linear inequalities. These inequalities are generated by considering all node subsets and requiring that the external flow associated with each subset be less or equal to the total capacity of the arcs exiting :from it. For a particularly appealing statement of...

We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs. We also give a t...

Representing discrete objects by polyhedral complexes, we can define all conceivable topological characteristics of points in discrete objects, namely those of vertices of polyhedral complexes. Such a topological characteristic is determined for each point by observing a configuration of object points in the 3 × 3 × 3 local point set of its neighbors. We study a topological characteristic such ...